A number is a mathematical object used to count, measure, order and label. The most basic examples are the natural numbers: 1, 2, 3, 4, 5, and so forth. Individual numbers can be represented in spoken or written language with number words, or with dedicated symbols called numerals; for example, "eleven" is a number word and "11" is the corresponding numeral. As only a limited list of symbols can be memorized, a numeral system is used to represent any number in an organized way. The most common representation is the Hindu–Arabic numeral system, a decimal system which can display any non-negative integer using a combination of ten Arabic numeral symbols called digits. Numerals can be used for counting (as with cardinal number of a collection or set), for labelling (as with telephone numbers), for ordering (as with serial numbers), and for codes (as with ISBNs). In common usage, however, a numeral is not clearly distinguished from the number that it represents.
In mathematics, the notion of a number has been extended over the centuries to include zero (0), negative numbers such as negative one (−1), rational numbers such as one half , real numbers such as the square root of 2 , and pi (), and complex numbers which extend the real numbers with a square root of (), and its combinations with real numbers by adding or subtracting its multiples. Calculations with numbers are done with arithmetical operations, the most familiar being addition, subtraction, multiplication, division, and exponentiation. Their study or usage is called arithmetic, a term which may also refer to number theory, the study of the properties of numbers.
Viewing the concept of zero as a number required a fundamental shift in philosophy, identifying nothingness with a value. During the 19th century, mathematicians began to develop the various systems now called algebraic structures, which share certain properties of numbers, and may be seen as extending the concept. Some algebraic structures are explicitly referred to as numbers (such as the -adic numbers and hypercomplex numbers) while others are not, but this is more a matter of convention than a mathematical distinction.
History
First use of numbers
Bones and other artifacts have been discovered with marks cut into them that many believe are tally marks. Some historians suggest that the Lebombo bone (dated about 43,000 years ago) and the Ishango bone (dated about 22,000 to 30,000 years ago) are the oldest arithmetic artifacts but this interpretation is disputed. These tally marks may have been used for counting elapsed time, such as numbers of days, lunar cycles or keeping records of quantities, such as of animals. A perceptual system for quantity thought to underlie numeracy, is shared with other species, a phylogenetic distribution suggesting it would have existed before the emergence of language.
A tallying system has no concept of place value (as in modern decimal notation), which limits its representation of large numbers. Nevertheless, tallying systems are considered the first kind of abstract numeral system.
The earliest unambiguous numbers in the archaeological record are the Mesopotamian base 60 (sexagesimal) system ( BC); place value emerged in the 3rd millennium BCE. The earliest known base 10 system dates to 3100 BC in Egypt. A Babylonian clay tablet dated to provides an estimate of the circumference of a circle to its diameter of = 3.125, possibly the oldest approximation of π.
Numerals
Numbers should be distinguished from numerals, the symbols used to represent numbers. The Egyptians invented the first ciphered numeral system, and the Greeks followed by mapping their counting numbers onto Ionian and Doric alphabets. (However, in 300 BC, Archimedes first demonstrated the use of a positional numeral system to display extremely large numbers in The Sand Reckoner.) Roman numerals, a system that used combinations of letters from the Roman alphabet, remained dominant in Europe until the spread of the Hindu–Arabic numeral system around the late 14th century, and the Hindu–Arabic numeral system remains the most common system for representing numbers in the world today. The key to the effectiveness of the system was the symbol for zero, which was developed by ancient Indian mathematicians around 500 AD.
Zero
The first known recorded use of zero as an integer dates to AD 628, and appeared in the Brāhmasphuṭasiddhānta, the main work of the Indian mathematician Brahmagupta. He is usually considered the first to formulate the mathematical concept of zero. Brahmagupta treated 0 as a number and discussed operations involving it, including division by zero. He gave rules of using zero with negative and positive numbers, such as "zero plus a positive number is a positive number, and a negative number plus zero is the negative number". By this time (the 7th century), the concept had clearly reached Cambodia in the form of Khmer numerals, and documentation shows the idea later spreading to China and the Islamic world. The concept began reaching Europe through Islamic sources around the year 1000.
There are other uses of zero before Brahmagupta, though the documentation is not as complete as it is in the Brāhmasphuṭasiddhānta. The earliest uses of zero was as simply a placeholder numeral in place-value systems, representing another number as was done by the Babylonians. Many ancient texts used 0, including Babylonian and Egyptian texts. Egyptians used the word nfr to denote zero balance in double entry accounting. Indian texts used a Sanskrit word or to refer to the concept of void. In mathematics texts this word often refers to the number zero. In a similar vein, Pāṇini (5th century BC) used the null (zero) operator in the Ashtadhyayi, an early example of an algebraic grammar for the Sanskrit language (also see Pingala).
Records show that the Ancient Greeks seemed unsure about the status of 0 as a number: they asked themselves "How can 'nothing' be something?" leading to interesting philosophical and, by the Medieval period, religious arguments about the nature and existence of 0 and the vacuum. The paradoxes of Zeno of Elea depend in part on the uncertain interpretation of 0. (The ancient Greeks even questioned whether was a number.)
The late Olmec people of south-central Mexico began to use a placeholder symbol for zero, a shell glyph, in the New World, by 38 BC. It would be the Maya who developed zero as a cardinal number, employing it in their numeral system and in the Maya calendar. Maya used a base 20 numerical system by combining a number of dots (base 5) with a number of bars (base 4). George I. Sánchez in 1961 reported a base 4, base 5 "finger" abacus.
By 130 AD, Ptolemy, influenced by Hipparchus and the Babylonians, was using a symbol for 0 (a small circle with a long overbar) within a sexagesimal numeral system otherwise using alphabetic Greek numerals. Because it was used alone, not as just a placeholder, this Hellenistic zero was the first documented use of a true zero in the Old World. In later Byzantine manuscripts of his Syntaxis Mathematica (Almagest), the Hellenistic zero had morphed into the Greek letter Omicron (otherwise meaning 70 in isopsephy).
A true zero was used in tables alongside Roman numerals by 525 (first known use by Dionysius Exiguus), but as a word, meaning nothing, not as a symbol. When division produced 0 as a remainder, , also meaning nothing, was used. These medieval zeros were used by all future medieval computists (calculators of Easter). An isolated use of their initial, N, was used in a table of Roman numerals by Bede or a colleague about 725, a true zero symbol.
Negative numbers
The abstract concept of negative numbers was recognized as early as 100–50 BC in China. The Nine Chapters on the Mathematical Art contains methods for finding the areas of figures; red rods were used to denote positive coefficients, black for negative. The first reference in a Western work was in the 3rd century AD in Greece. Diophantus referred to the equation equivalent to (the solution is negative) in Arithmetica, saying that the equation gave an absurd result.
During the 600s, negative numbers were in use in India to represent debts. Diophantus' previous reference was discussed more explicitly by Indian mathematician Brahmagupta, in Brāhmasphuṭasiddhānta in 628, who used negative numbers to produce the general form quadratic formula that remains in use today. However, in the 12th century in India, Bhaskara gives negative roots for quadratic equations but says the negative value "is in this case not to be taken, for it is inadequate; people do not approve of negative roots".
European mathematicians, for the most part, resisted the concept of negative numbers until the 17th century, although Fibonacci allowed negative solutions in financial problems where they could be interpreted as debts (chapter 13 of , 1202) and later as losses (in ). René Descartes called them false roots as they cropped up in algebraic polynomials yet he found a way to swap true roots and false roots as well. At the same time, the Chinese were indicating negative numbers by drawing a diagonal stroke through the right-most non-zero digit of the corresponding positive number's numeral. An early European experimenter with negative numbers was Nicolas Chuquet during the 15th century. He used them as exponents, but referred to them as "absurd numbers".
As recently as the 18th century, it was common practice to ignore any negative results returned by equations on the assumption that they were meaningless.
Rational numbers
It is likely that the concept of fractional numbers dates to prehistoric times. The Ancient Egyptians used their Egyptian fraction notation for rational numbers in mathematical texts such as the Rhind Mathematical Papyrus and the Kahun Papyrus. The Rhind Papyrus includes an example of deriving the area of a circle from its diameter, which yields an estimate of π as ≈ 3.16049.... Classical Greek and Indian mathematicians made studies of the theory of rational numbers, as part of the general study of number theory. A particularly influential example of these is Euclid's Elements, dating to roughly 300 BC. Of the Indian texts, the most relevant is the Sthananga Sutra, which also covers number theory as part of a general study of mathematics.
The concept of decimal fractions is closely linked with decimal place-value notation; the two seem to have developed in tandem. For example, it is common for the Jain math sutra to include calculations of decimal-fraction approximations to pi or the square root of 2. Similarly, Babylonian math texts used sexagesimal (base 60) fractions.
Real numbers and irrational numbers
The Babylonians, as early as 1800 BCE, demonstrated numerical approximations of irrational quantities such as √2 on clay tablets, with an accuracy analogous to six decimal places, as in the tablet YBC 7289. These values were primarily used for practical calculations in geometry and land measurement. There were practical approximations of irrational numbers in the Indian Shulba Sutras composed between 800 and 500 BC.
The first existence proofs of irrational numbers is usually attributed to Pythagoras, more specifically to the Pythagorean Hippasus, who produced a (most likely geometrical) proof of the irrationality of the square root of 2. The story goes that Hippasus discovered irrational numbers when trying to represent the square root of 2 as a fraction. However, Pythagoras believed in the absoluteness of numbers. He could not disprove the existence of irrational numbers, or accept them, so according to legend, he sentenced Hippasus to death by drowning, to impede the spread of this unsettling news.